MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eldm Structured version   Unicode version

Theorem eldm 5198
Description: Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)
Hypothesis
Ref Expression
eldm.1  |-  A  e. 
_V
Assertion
Ref Expression
eldm  |-  ( A  e.  dom  B  <->  E. y  A B y )
Distinct variable groups:    y, A    y, B

Proof of Theorem eldm
StepHypRef Expression
1 eldm.1 . 2  |-  A  e. 
_V
2 eldmg 5196 . 2  |-  ( A  e.  _V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
31, 2ax-mp 5 1  |-  ( A  e.  dom  B  <->  E. y  A B y )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   E.wex 1596    e. wcel 1767   _Vcvv 3113   class class class wbr 4447   dom cdm 4999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-dm 5009
This theorem is referenced by:  dmi  5215  dmcoss  5260  dmcosseq  5262  dminss  5418  dmsnn0  5471  dffun7  5612  dffun8  5613  fnres  5695  opabiota  5928  fndmdif  5983  dff3  6032  frxp  6890  reldmtpos  6960  dmtpos  6964  aceq3lem  8497  axdc2lem  8824  axdclem2  8896  fpwwe2lem12  9015  nqerf  9304  shftdm  12863  xpsfrnel2  14816  bcthlem4  21501  dchrisumlem3  23404  eupath  24657  fundmpss  28773  elfix  29130  fnsingle  29146  fnimage  29156  funpartlem  29169  dfrdg4  29177  prtlem16  30214
  Copyright terms: Public domain W3C validator